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International Conference on Communications

An analysis into the loopy belief propagation algorithm over short cycles

Nithin Raveendran, Shayan Garani Srinivasa

Published Jun 1, 2014• 5 citations
Publisher

Abstract

We investigate into the loopy belief propagation algorithm for binary low density parity check (LDPC) codes having cycles of small girth. Independence assumption among messages passed, assumed reasonable in all configurations of graphs, fails the most in graphical structures with short cycles. We investigate into this limitation and propose a modified algorithm, by considering dependency in the probability domain. This improves the performance of decoding over such graphs when compared to the original message passing algorithm at higher signal-to-noise ratio (SNR), thereby, yielding lower error floors.

DOI
10.1109/icc.2014.6883618

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We investigate into the loopy belief propagation algorithm for binary low density parity check (LDPC) codes having cycles of small girth. Independence assumption among messages passed, assumed reasonable in all configurations of graphs, fails the most in graphical structures with short cycles. We investigate into this limitation and propose a modified algorithm, by considering dependency in the probability domain. This improves the performance of decoding over such graphs when compared to the original message passing algorithm at higher signal-to-noise ratio (SNR), thereby, yielding lower error floors.
We investigate into the limitations of the sum-product algorithm for binary low density parity check (LDPC) codes having isolated short cycles. Independence assumption among messages passed, assumed reasonable in all configurations of graphs, fails the most in graphical structures with short cycles. This research work is a step forward towards understanding the effect of short cycles on error floors of the sum-product algorithm. We propose a modified sum-product algorithm by considering the statistical dependency of the messages passed in a cycle of length 4. We also formulate a modified algorithm in the log domain which eliminates the numerical instability and precision issues associated with the probability domain. Simulation results show a signal to noise ratio (SNR) improvement for the modified sum-product algorithm compared to the original algorithm. This suggests that dependency among messages improves the decisions and successfully mitigates the effects of length-4 cycles in the Tanner graph. The improvement is significant at high SNR region, suggesting a possible cause to the error floor effects on such graphs. Using density evolution techniques, we analysed the modified decoding algorithm. The threshold computed for the modified algorithm is higher than the threshold computed for the sum-product algorithm, validating the observed simulation results. We also prove that the conditional entropy of a codeword given the estimate obtained using the modified algorithm is lower compared to using the original sum-product algorithm.
We present a modified belief propagation (BP) algorithm for decoding low density parity check codes having graphs with short cycles. The modified algorithm in log domain is superior in terms of numerical stability, precision, computational complexity and ease of implementation when compared to the algorithm in the probability domain. Simulation results show improvement in decoding performance for the modified BP compared to the original algorithm. The modified approach is also generalized for graphs with isolated cycles of arbitrary length by considering the statistical dependency among messages passed in such cycles. © 2019 International Foundation for Telemetering. All rights reserved.
We investigate into the limitations of the sum-product algorithm in the probability domain over graphs with isolated short cycles. By considering the statistical dependency of messages passed in a cycle of length 4, we modify the update equations for the beliefs at the variable and check nodes. We highlight an approximate log domain algebra for the modified variable node update to ensure numerical stability. At higher signal-to-noise ratios (SNR), the performance of decoding over graphs with isolated short cycles using the modified algorithm is improved compared to the original message passing algorithm (MPA).
In this paper, a modification to belief propagation (BP) decoding algorithm is proposed, which is based on extracting the prior messages of each variable node to help the BP decoding, and is particularly effective for low-density parity-check (LDPC) codes with short cycles, where the existence of cycles makes the original BP algorithm perform suboptimal.The proposed algorithm, referred to as "employing the positive effects of the feedback messages (EPEFM)", extracts the positive effects of the feedback messages and then makes use of them as prior messages to assist the decoding of the BP algorithm.Simulation results confirm the effectiveness of our proposed algorithm, which improves the performance in high signal-to-noise-ratio (SNR) region without loss in low SNR region.
Error control coding is an essential part of modern communications systems. LDPC codes have been demonstrated to offer performance near the fundamental limits of channels corrupted by random noise. Optimal maximum likelihood decoding of LDPC codes is too complex to be practically useful even at short block lengths and so a graph-based message passing decoder known as the belief propagation algorithm is used instead. In fact, on graphs without closed paths known as cycles the iterative message passing decoding is known to be optimal and may converge in a single iteration, although identifying the message update schedule which allows single-iteration convergence is not trivial. At finite block lengths graphs without cycles have poor minimum distance properties and perform poorly even under optimal decoding. LDPC codes with large block length have been demonstrated to offer performance close to that predicted for codes of infinite length, as the cycles present in the graph are quite long. In this thesis, LDPC codes of shorter length are considered as they offer advantages in terms of latency and complexity, at the cost of performance degradation from the increased number of short cycles in the graph. For these shorter LDPC codes, the problems considered are: \n \nFirst, improved construction of structured and unstructured LDPC code graphs of short length with a view to reducing the harmful effects of the cycles on error rate performance, based on knowledge of the decoding process. Structured code graphs are particularly interesting as they allow benefits in encoding and decoding complexity and speed. Secondly, the design and construction of LDPC codes for the block fading channel, a particularly challenging scenario from the point of view of error control code design. Both established and novel classes of codes for the channel are considered. Finally the decoding of LDPC codes by the belief propagation algorithm is considered, in particular the scheduling of messages passed in the iterative decoder. A knowledge-aided approach is developed based on message reliabilities and residuals to allow fast convergence and significant improvements in error rate performance.
In this paper, we propose a binary message-passing algorithm for decoding low-density parity-check (LDPC) codes. The algorithm substantially improves the performance of purely hard-decision iterative algorithms with a small increase in the memory requirements and the computational complexity. We associate a reliability value to each nonzero element of the code's parity-check matrix, and differentially modify this value in each iteration based on the sum of the extrinsic binary messages from the check nodes. For the tested random and finitegeometry LDPC codes, the proposed algorithm can perform as close as about 1 dB and 0.5 dB to belief propagation (BP) at the error rates of interest, respectively. This is while, unlike BP, the algorithm does not require the estimation of channel signal to noise ratio. Low memory and computational requirements and binary message-passing make the proposed algorithm attractive for high-speed low-power applications.